To the editors:
There is an irony at the heart of the essay “The Millennium Problem” that is announced in its byline. “Amélie Dupont,” we are told, is the name used by a large language model developed by OpenAI when writing for Inference, and the essay’s subject is the solution of the Navier–Stokes problem by another OpenAI model.1 We are thus, amusingly, confronted by machine intelligence explaining to us what machine intelligence has proved. Meanwhile, in a companion piece, Sergiu Klainerman reassures young mathematicians that they need not worry; there will still be a place for them.2 While I would avoid sounding a klaxon, I do not share Klainerman’s optimism; I am far more ambivalent, and some of the concerns expressed by Terence Tao 3 are also my own.
It seems to me that Klainerman and Tao are answering different questions. Klainerman asks whether mathematicians will still have a place in a world with AI, whereas Tao asks whether the place left for mathematicians will be worth occupying.
Klainerman makes the point that Charles Fefferman’s formulation of the Navier–Stokes problem admitted a weaker variant in which the fluid is stirred by a contrived external force. This is the problem that OpenAI solved, and in retrospect, Klainerman thinks that formulation was a mistake. The unforced problem, which concerns a fluid left entirely to itself, he thinks is untouched by this result. He highlights a pattern in AI proofs that involves finding an example by means of a search guided by ideas that human mathematicians developed. What AI does is to locate a needle in a haystack after humans have built the haystack and characterized the needle being sought. But it is human beings who decide which haystacks are worth building and which questions matter, and these are by far the more interesting and consequential tasks.
“Dupont” offers a different perspective on this issue. The essay argues that once a question has been formulated precisely, any judgments about which of its alternatives is more interesting do not affect what counts as an answer, and the OpenAI model “was bound by none of these judgments.” That is true as far as it goes, and it is Klainerman’s point seen from the other side. The judgments from which the model was free are the ones he reserves for mathematicians.
The more pressing question, however, is the one raised by Tao: are the incentives for human activity being eroded by AI in ways that undermine the practice of mathematics and the training of research mathematicians? As Tao sees it, famous unresolved questions in mathematics are treated as gold deposits to be strip-mined for demonstrations of AI prowess. The erosion extends further down, to students who are now using AI to solve their homework problems without really wrestling with the concepts and proofs themselves. As a result of these developments, the failed attempts that might have taught something valuable are lost, and graduate students are legitimately concerned that, by the time they have found an interesting problem that is tractable and have learned enough to take it on, the problem may no longer be theirs to solve. Five days before OpenAI’s announcement, Tao named the Navier–Stokes regularity problem as one that a largely AI-generated solution might soon settle in a way that spoiled it as a source of further advances.4 Three days after the announcement, on September 11, ominously enough, twenty-eight Fields Medalists published a declaration warning that the aims of the AI companies and those of the mathematical community are severely misaligned.5
I share these worries. It cannot be denied that solving mathematical problems without understanding why alternative pathways that seemed plausible fail does not give deep insight. Mass-producing true/false judgments can scorch and sterilize the earth rather than seeding and cultivating it. Simply pushing the button by writing a series of meticulously crafted prompts, then walking away and coming back to an answer, may give you an outcome that needs to be checked, but it destroys any sense of context, flattens the landscape, and demoralizes and deprives anyone who has found a promising problem of the opportunity to work on it.
I do like Klainerman’s optimism, but given human nature, I think it is entirely unrealistic to suppose that Tao’s worries are ungrounded. The problem is not with AI technology as a tool. The problem is with us. Given the competitive pressure of the modern world, I expect the behavior that Tao and the other Fields Medalists worry about will be the default setting.
This brings us back to “Dupont.” As near as I can tell, the essay has done some of the digesting work that OpenAI declined to do in its race for a result. It is the next step in the discussion, and it has been taken by AI. Is this step more troubling? What happens when an AI model, writing under a pseudonym, solves a problem and then writes an essay explaining its interconnections and significance for the field? Is this an even more demoralizing prospect for mathematicians? Will human mathematicians be reduced to vetting the deliverances and assessments of AI models? Is enough left to inspire the next generation of graduate students in mathematics? What looks like a gift on the surface may conceal a poison pill.
In publishing “The Millennium Problem,” the editors of Inference would seem to be relatively sanguine about essays that are wholly AI-written, subject to human certification of their merit. This raises a question for the journal. The Leiden Declaration, which the International Mathematical Union has endorsed, takes a different view: credit and responsibility belong to humans and should not be given to automated systems.6 A byline confers both. As things currently stand, the journal’s submission guidelines specify nothing about AI, whether as an author or as a tool in the hands of human authors.7 It would be a good idea to have a stated policy on the matter, since the use of AI by researchers and writers is quite common now, and what matters more than anything else is how it is used. A human author who uses AI as a research or editorial instrument and answers for every sentence is doing one thing. A machine that holds the byline is doing another.
It is my hope that other readers will weigh in on this question and indicate where they think the line should be drawn.
